Statistics Question

Question

Let X have a U(0, 1) distribution, with cumulative distribution function F(·).

Prove that Y = − ln(1 − F(X)) has an exponential distribution and give its parameter. Then, derive the moment generating function (mgf) of Y.

Answer 1

U(0,1) means that dF/dx = 1 (0 ≤ x ≤ 1), 0 (else) and F(x) = 0 (x < 0), x (0 ≤ x ≤ 1), 1 (1 < x).

The general formula for p(y = g(x)) is ∫₀¹δ(y – g(x))dF(x) (as can be seen by y-integration) so,

p(y) = ∫₀¹δ(y + ln(1 – F(x)))dF(x)

= ∫₀¹δ(y + ln(1 – F))dF

= ∫₀¹δ(y + ln v)dv

= ∫₀^∞δ(y – w)exp(–w)dw

= exp(–y) (0 ≤ y), 0 (y < 0)

that is the exponential distribution with parameter λ = 1.

mgf(t) = ∫₀^∞exp(ty)exp(–y)dy

= 1/(1 – t) for t < 1.